Lower semicontinuity and relaxation of linear-growth integral functionals under PDE constraints-Reference-Cited by-同舟云学术

Lower semicontinuity and relaxation of linear-growth integral functionals under PDE constraints

Published:2018-01-11 Issue:3 Volume:13 Page:219-255
ISSN:1864-8266
Container-title:Advances in Calculus of Variations
language:en
Short-container-title:

Author:

Arroyo-Rabasa Adolfo¹^ORCID,De Philippis Guido²,Rindler Filip³^ORCID

Affiliation:

1. Mathematisches Institut , Universität Leipzig , Augustusplatz 10, 04109 Leipzig , Germany

2. Scuola Internazionale Superiore di Studi Avanzati , Via Bonomea 265, 34136 Trieste , Italy

3. Mathematics Institute , University of Warwick , Coventry CV4 7AL , United Kingdom

Abstract

Abstract We show general lower semicontinuity and relaxation theorems for linear-growth integral functionals defined on vector measures that satisfy linear PDE side constraints (of arbitrary order). These results generalize several known lower semicontinuity and relaxation theorems for BV, BD, and for more general first-order linear PDE side constrains. Our proofs are based on recent progress in the understanding of singularities of measure solutions to linear PDEs and of the generalized convexity notions corresponding to these PDE constraints.

Funder

Engineering and Physical Sciences Research Council

Publisher

Walter de Gruyter GmbH

Subject

Applied Mathematics,Analysis

Link

https://www.degruyter.com/document/doi/10.1515/acv-2017-0003/pdf

Reference32 articles.

1. J. J. Alibert and G. Bouchitté, Non-uniform integrability and generalized Young measures, J. Convex Anal. 4 (1997), no. 1, 129–147.

2. L. Ambrosio and G. Dal Maso, On the relaxation in BV⁢(Ω;𝐑m){{\rm BV}(\Omega;\mathbf{R}^{m})} of quasi-convex integrals, J. Funct. Anal. 109 (1992), no. 1, 76–97.

3. L. Ambrosio, N. Fusco and D. Pallara, Functions of Bounded Variation and Free Discontinuity Problems, Oxford Math. Monogr., The Clarendon Press, New York, 2000.

4. A. Arroyo-Rabasa, Relaxation and optimization for linear-growth convex integral functionals under PDE constraints, J. Funct. Anal. 273 (2017), no. 7, 2388–2427.

5. M. Baía, M. Chermisi, J. Matias and P. M. Santos, Lower semicontinuity and relaxation of signed functionals with linear growth in the context of 𝒜{\mathscr{A}}-quasiconvexity, Calc. Var. Partial Differential Equations 47 (2013), no. 3–4, 465–498.

Cited by 24 articles. 订阅此论文施引文献订阅此论文施引文献，注册后可以免费订阅5篇论文的施引文献，订阅后可以查看论文全部施引文献

1. Measure-Valued Structured Deformations;Journal of Nonlinear Science;2024-09-03

2. A Lebesgue–Lusin property for linear operators of first and second order;Proceedings of the Royal Society of Edinburgh: Section A Mathematics;2023-11-06

3. Fine properties of symmetric and positive matrix fields with bounded divergence;Advances in Mathematics;2023-08

4. Which Measure-Valued Solutions of the Monoatomic Gas Equations are Generated by Weak Solutions?;Archive for Rational Mechanics and Analysis;2023-06-05

5. On Scaling Properties for Two-State Problems and for a Singularly Perturbed $T_{3}$ Structure;Acta Applicandae Mathematicae;2023-03-17